Molecular Weight Development during Simultaneous Chain Scission, Long‐Chain Branching and Crosslinking, 1

Abstract

A general matrix formula is proposed for the weight‐average molecular weights of the polymer systems formed through simultaneous scission, branching and crosslinking of N types of chains, assuming the chain connection statistics are Markovian. For the polymerization systems in which chains are generated consecutively, such as for free‐radical polymerization, the present theory can be applied by increasing the number of chain types N to infinity, by considering the chains formed at different times as different types of chains. The gel point determination reduces to the eigenvalue problem and the present theory extends the classical gelation theory to non‐random, history‐dependent reaction systems. From the mathematical point of view, this theory is capable of describing complex molecular build‐up processes through end‐linking, T‐ and H‐shaped chain connections, irrespective of reaction/reactor types used.

Markovian Approach to Self‐Condensing Vinyl Polymerization: Distributions of Molecular Weights, Degrees of Branching, and Molecular Dimensions

“…The random sampling technique is used for developing a model to describe the SCVP process. With the random sampling technique on a weight basis, a polymer molecule is selected by choosing one monomeric unit randomly . The weight‐average chain length is the expected size of the polymer molecule so selected, which can be obtained in a matrix formula .…”

“…In this paper, the random sampling technique is applied on a weight basis, i.e., each polymer molecule is selected by choosing one monomeric unit randomly . With this method, the analytic expression for the weight‐average chain length in a matrix formula can be obtained . When the concept of random sampling technique is applied to the MC simulation, one can investigate the complex molecular structure one by one and very detailed information can be obtained.…”

Polymerization Processes, 1. Fundamentals

“…Assuming that the effects of the size and/or structure dependent kinetics are negligible, the random sampling technique introduced in this article makes it possible to formulate the massaverage chain length in a matrix formula for nonrandom branching and/or crosslinking, generally represented by [87][88][89][90]: P non-randomness ¼ P non-randomness;0 þ mFðI À MÞ À1 s ð184Þ…”

Molecular Weight Development during Simultaneous Chain Scission, Long‐Chain Branching and Crosslinking, 2

“…The polymerization system considered in this article is the same as that described in Part 1 of this series,1 consisting of the following elementary reactions. Initiation (rate in $[{\rm mol}/{\rm l} \cdot {\rm s}],\;R_I = 2k_d f[I])$ , propagation $(R_p = k_p [M][R^ \bullet ])$ , chain transfer to small molecules such as monomer M , solvent S and chain transfer agents T $(R_f = (k_{fm} [M] + k_{fs} [S] + k_{fT} [T])[R^ \bullet ])$ .…”

“…In Part 1 of this series,1 a general matrix formula describing the weight‐average molecular weight of the polymer systems formed through the end‐linking, T‐ and H‐shaped chain connections is proposed. This article shows how the matrix formula is applied to free‐radical (co)polymerization systems.…”